Greenland, need help to travel to....

There are no direct flights from the US. I think connecting through Iceland or Denmark is the way to go. Google "travel to Greenland" and you'll see a ton of sites specifically about how to travel to Greenland.

There are direct flights to Nuuk (the Capital) from Reykjavik, so you may be able to string together an itinerary by flying Icelandair.
Once there you can book day trips to see the glaciers.
Greenland is one of those remote places that probably requires a formal tour to make the best use of your time. I see a few on Road Scholar that look interesting. Though not cheap it is, as you say, a bucket list item so splurging a little to see as much as possible with a knowledgeable guide might be justified in this case.
Historically, April is the sunniest month (though still bloody cold) while July is the warmest with average daily temps in the 50's.

Yes, I agree, a tour (preferably an expedition type tour) would be your best way to see Greenland.
There are very few paved roads, and travel between cities is typically by boat or plane.

We went on a National Geographic/Lindblad expedition, which included both Iceland and Greenland. We visited the lower western portion of Greenland and LOVED the experience. We had amazing insight into the territory (of Denmark), as we had the previous female prime minister along as our global perspectives speaker. To see Greenland is fabulous, but to learn about the culture from excellent resources adds so much to the educational aspect of the trip........one comes awy with a much greater understanding of the beauty, the history, current life, and potential opportunities/challenges for the future.

National Geo no longer offers the specific itinerary we took (it was a compressed trip of both locations), but they do offer similar other itineraries. They are incredible and worth every penny.

Famous for running the Norwegian coastal route, https://www.hurtigruten.com/find-your-voyage/ also has (expensive) cruises from Canada and/or Iceland to Greenland. Most departures are in mid-summer for warmer weather.

May is considered neither winter or summer in Greenland. Because Greenland is on the north hemisphere as San Francisco is, May is considered spring. My educated guess is that cheaper time of the year to get there would be November, December, January, February.

Consider combining with Iceland, especially if you can get a good IcelandAir flight to Iceland. I met a couple of tourists in Iceland that took a flight over to Greenland from Iceland and really enjoyed it. Reykjavik in Iceland will be a real nice contrast to Greenland which I expect to be largely rural/very small town. Just be warned that Iceland is expensive, but totally worth seeing in combination with your top interest.

You can also do a day tour from Iceland to Greenland.

Road Scholar has a number of tours for Greenland and Iceland with even free airfare on some.

https://www.roadscholar.org/find-an-adventure/?query=greenland

Take a cruise. We visited Iceland while on a cruise and loved it. Take the Golden Circle tour.